Rates of convergence for the three state contact process in one dimension

The basic contact process with parameter $\mu$ altered so that infections of sites that have not been previously infected occur at rate proportional to $\lambda$ instead is considered. Emergence of an infinite epidemic starting out from a single infected site is not possible for $\mu$ less than the contact process’ critical value, whereas it is possible for $\mu$ greater than that value. In the former case the space and time infected regions are shown to decay exponentially; in the latter case and for $\lambda$ greater than $\mu$, the ratio of the endmost infected site’s velocity to that of the contact process is shown to be at most $\lambda \mathrm{/}\mu$.